WARM UP 1.In the expression, the number 7 is called the _________. 2.In the expression, the number 5 is called the _________. 3.The entire expression is called an ___________ 4.Write the following as single exponential expressions. 1.Write the expression without parenthesis. 2.Explain the meaning of base exponent exponential expression

PROPERTIES OF LOGARITHMS

OBJECTIVES  Apply the basic properties of logarithms  Simplify exponential and logarithmic expressions  Learn the properties of base-10 logarithms.

KEY TERMS & CONCEPTS  Base-10 logarithm  Logarithm  Logarithm of a product  Logarithm of a quotient  Logarithm of a power  Logarithm is an exponent

INTRODUCTION Any positive number can be written as power of 10. For instance: The exponents …, …, and … are called the base-10 logarithms of 3, 5, and 15, respectively. Your calculator is programmed to calculate these numbers when you press LOG.

DEFINITION & PROPERTIES OF BASE-10 PROPERTIES To gain confidence in the meaning of logarithm, press log 3 on your calculator. You will get: Then, without rounding, raise 10 to this power. You will get The powers of 10 have the normal properties of exponentiation. For instance: You can check by calculator that really does equal 15. Add the exponents; keep the same base.

DEFINITION & PROPERTIES OF BASE-10 PROPERTIES From this example you can infer that logarithms have the same properties as exponents. This is not surprising, because logarithms are exponents. For instance log (3  5) = log 3 + log 5 From the values given earlier, you can also show that The log of a product equals the sum of the logs of the factors. The log of a quotient. This property is reasonable because you divide powers of equal bases by subtracting the exponents.

DEFINITION & PROPERTIES OF BASE-10 PROPERTIES Because a power can be written as a product, you can find the logarithm of a power like this: log 3 = log (3  3  3  3) = log 3 + log 3 + log 3 + log 3 = 4 log 3 Combine like terms. The logarithm of a power equal the exponent of that power, times the logarithm of the base. To verify this result, observe that 3 = 81. press 4 log 3 on your calculator, and get … Then press log 81. You get the same answer, …

DEFINITIONS & PROPERTIES: BASE-10 LOGARITHMS DEFINITION log x = y if an only if 10 = x Verbally: log x is the exponent in the power of 10 that gives x PROPERTIES Log of a Product. log (xy) = log x + log y Verbally: the log of a product = sum of the logs of the factors Log of a Quotient log x/y = log x – log y Verbally: the log of a quotient = log of numerator minus log of denominator Log of a Power log x = y  log x Verbally: log of a power = exponent times log of the base.

DEFINITIONS The reason for the name logarithm is historical. Before there were calculators, base-10 logarithms, calculated approximately using infinite series, were recorded in table. Products with many factors, such as could then be calculated by adding their logarithms (exponents) column-wise in one step rather than by tediously multiplying several pairs of numbers. The word logarithm actually comes from the Greek word logos, which here means “ratio”, and “arithmos”, which means number. (357)(4.367)(22.4)(3.142) The most important thing to remember about logarithm is that: A LOGARITHM IS AN EXPONENT

EXAMPLE 1  Examples 1 and 2 show you how to verify that logarithm is an exponent. Find x if. Verify your solution numerically Solution: By definition, the logarithm is the exponent of 10. So x = Check: = By calculator. Do not round. log = which checks..

EXAMPLE 2 Find x if. Verify your solution numerically Solution: By definition, the exponent of 10 is the logarithm of Check: By calculator. Do not round. x = log = which checks..

EXAMPLE 3  Examples 3, 4 and 5 show you how to verify that numerically the three properties of logarithms. Show numerically that log (7  9) = log 7 + log 9. Explain how this property agrees with the definition of logarithms. Solution: log (7  9) = log 63 = … This equality agrees with the definition because By calculator. Do not round. log 7 + log 9 = … … log (7  9) = log 7 + log 9 7 · 9 = ( )( )

SOLUTION CONT. This equality agrees with the definition because Calculate without rounding. 7 · 9 = ( )( ) = = log (7 · 9) = The logarithm is the exponent of 10.

EXAMPLE 4 Show numerically that. Explain how this property agrees with the definition of logarithm. Solution: Calculate without rounding. log 51/17 = log 3 = log 51 − log 17 = − = log = log 51 – log 17

SOLUTION CONT. This equality agrees with the definition because Subtract the exponents. Keep the same base. The logarithm is the exponent of 10. =

EXAMPLE 5 Show numerically that 5 = 3 log 5. Explain how this property can be derived from the log of a product property. Solution: Calculate without rounding. log 5 3 = log 125 = … 3 log 5 = 3 · = log 5 = 3 log 5

SOLUTION CONT. This equality agrees with the definition because The log of a product equals the sum of the logs of the factors. Combine like terms. log 5 3 = log (5 · 5 · 5) = log 5 + log 5 + log 5 = 3 log 5

EXAMPLE 6 Example 6 shows you how to prove algebraically that the logarithm of the product of two numbers equals the sum of the logarithms of the factors. Prove algebraically that log xy = log x + log y. Proof: Let c = log x and let d = log y Then 10c = x and 10 d = y By definition of logarithm.. Multiply x times y. Add the exponents. Keep the same base.. The logarithm is the exponent of 10. Substitute for c and d..

EXAMPLE 6 Example 7 shows you how to use the properties of logarithms to simplify expressions that contain logarithms Use the properties of logarithms to find the number that goes in the blank: log 3 + log 7 – log 5 = log ______. Check you answer numerically. Solution: log 3 + log 7 – log 5 = log = log goes in the blank By calculator. Check: log 3 + log 7 – log 5 = log 4.2 = log 3 + log 7 − log 5 = log 4.2

CH. 7.4 Homework Textbook pg. 319 # 2-44 Every other even & Journal Question #47